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6. Longitudinal Maneuvering Stability
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For Training Purposes Only
Maneuvering Stability
Longitudinal maneuvering stability is really just
static stability with an additional factor: pitch
rate. An aircraft in accelerated (curved)
flight—whether pulling up, pushing over, or
turning—has a pitch rate. Figure 1 shows the
simple case of an aircraft in a pull-up. The
aircraft pitches about its c.g. The tail sweeps
along behind, on its arm, lT. The tail’s motion
creates a change in its relative wind and thus in
tail angle of attack, αT. The change in tail angle
of attack due to pitch rate produces an opposing
pitching moment, known as pitch damping.
The change in tail angle of attack, ΔαΤ, due to
pitch rate is shown in the formula below, where
q is pitch rate in radians per second (one radian
equals 57.3o; and 0.1 radian/second is
approximately 1 RPM). lT is the distance
between aircraft c.g. and the aerodynamic center
of the tail. VT is the velocity of the tail (taken
tangentially to the aircraft’s flight path).
Thus the faster you pitch, and/or the farther back
your tail, the greater the change in αΤ, but it’s all
inversely proportional to speed, VT, as the
formula shows.
Figure 1
Pitch Damping
Δα T =
ql T
VT
VT
Tail’s
aerodynamic
center
Tail arm, lT
cg
Pitch rate, q
ΔαΤ, change in tail angle of
attack
VT
qlT
(pitch rate
times arm)
Figure 2
Stick forceper-g gradient
versus c.g.
Stick
Force
Stick
Force
Turning
flight
At aft c.g.
At forward
c.g.
Wings-level pull up
1g
+g
+g
The actual tail angle of attack will also depend
on the increased downwash produced by the
wing as its lift coefficient rises in the pull-up,
and a proper formula would take that into
account.
Because of pitch damping, an aircraft is actually
more stable in maneuvering flight than in flight
at 1-g. Remember, we assess stability in terms of
the force needed to displace the aircraft from
equilibrium (trim). We assess static stability in
terms of the push or pull on the stick necessary
to change the coefficient of lift, CL, and to
produce airspeeds different than trim, while
flying at 1-g. In maneuvering flight at more than
1-g, pitch damping increases the stick force we
have to apply to displace the aircraft from
equilibrium. How rapidly stick forces will
increase as we increase g depends on the
maneuvering characteristics for which the
aircraft was designed, and its c.g. location. We
can examine an aircraft’s stick-fixed (elevator
position-per-g) and the really more
germane—since it’s what the pilot feels—stickfree (stick force-per-g) maneuvering
characteristics.
Figure 2 shows how the gradient, or slope, of
stick force-per-g depends on the location of the
aircraft c.g. Forward c.g. increases an aircraft’s
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6.1
Longitudinal Maneuvering Stability
maneuvering stability, and therefore stick forces
become heavier. As you move the c.g. back,
stick forces required to pull g go down. (The
stick position-per-g curve behaves similarly. As
c.g. moves aft, the deflection required to pull g
goes down.)
Stick force-per-g also varies directly with wing
loading (aircraft weight divided by wing area).
Highly wing loaded aircraft may need the help of
a powered control system to keep forces in
check. Raising the wing loading has the same
effect as moving the c.g. forward.
Stick force-per-g is a particularly important
parameter and one of the basic handling quality
differences between aircraft designed for
different missions. When we maneuver an
aircraft, we tend to evaluate its response in terms
of the force we apply to the stick rather than the
change in stick position. We know the stick has
returned to the equilibrium trim position, for
example, when the force disappears (at least
ideally—friction and other factors can get in the
way). And when we move the c.g. well aft in an
aircraft—or take that first aerobatic flight—it’s
the reduction in stick forces we probably notice
first.
Fighters and aerobatic aircraft require lower
forces-per-g than do normal or transport category
aircraft because their g envelopes are wider and
the total stick force necessary at high g would
otherwise be too great for the pilot to sustain. So
a fighter operating at up to 9-g or more needs a
shallower force-per-g gradient than a transport
expected to operate at no greater than the 1.5-g
approximately required for a 45-degree-bank
level turn. The fighter’s shallow force-per-g
gradient would be devastating in a transport
because the pilot could easily overstress the
aircraft. The transport’s steeper gradient would
have the fighter pilot pulling with both hands
while pushing on the instrument panel with his
feet.
The importance of stick force-per-g in fighters
became apparent during World War II. It was
decided that the upper limit should be about 8
lb/g to keep the pilot from tiring in a fight, with a
lower limit of 3 lb/g to prevent overstressing the
aircraft and losing by default.
Overstress is the big worry; so FAR Part 23.155
specifies the minimum total control force
necessary to reach an aircraft’s positive limit
6.2
maneuvering load factor (g limit). It’s based on
aircraft weight and the type of control. For wheel
controls the minimum force has to be at least 1%
of the aircraft’s maximum weight or 20 pounds,
whichever is greater, but doesn’t have to exceed
50 pounds. For stick controls, minimum force for
maximum g has to be at least max weight/140, or
15 pounds, whichever is greater, but doesn’t
have to exceed 35 pounds.
To figure out what that would mean in terms of
required average minimum control-force-per-g
gradient, you can take the design load limit of
the airplane (6-g’s for our trainers), subtract 1-g
to get the maximum g-load actually applied, and
then divide that into the minimum total force
required by regulation. For the Air Wolf (6-g’s
and 2900 lbs. maximum aerobatic weight):
2900
140
5
= 4.1 lb/g minimum allowable stick force
A Cessna 172’s yoke force is greater than 20lb/g.
A wings-level, 1.7-g pull-up in a Boeing 777
requires 135 pounds. The Boeing is certified
under FAR Part 25, which actually doesn’t
contain sustained maneuvering control force
requirements.
The FARs doesn’t specify maximum stick-forceper-g, but the military does, depending on the
type of aircraft.
Aircraft with shallow stick force-per-g gradients
can feel dramatically sensitive if your muscle
memory expects greater forces. Even
experienced aerobatic pilots stepping up to
higher performance aerobatic aircraft usually
find themselves pulling too hard, detaching the
boundary layer, and buffeting the
aircraft—especially in the excitement of
aerobatic competition. This is seen from the
ground as an abrupt flattening in the arc of a
loop, and from the cockpit as a sudden g-break.
But after one becomes accustomed to those
shallow gradients, the lower performance
aerobatic aircraft one trained in can seem
disagreeably reluctant to maneuver. The physical
effort now feels out of proportion to the result.
On the other hand, pilots of early swept wing
fighters had to worry about “g-limit overshoot”
because of the forward shift in the center of lift
as the tips began to stall. The F-86E Sabre
Aircraft Operating Instructions cautioned pilots
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Longitudinal Maneuvering Stability
against “A basic characteristic toward
longitudinal instability under conditions of high
load factor, which … results in a tendency to
automatically increase the rate of turn or pull-up
to the point where the limit load factor may be
exceeded.” Fortunately, this was preceded by
lots of warning buffet.
As noted, pitch damping depends on pitch rate.
Pitch rate depends not just on how hard you pull,
but also on the kind of maneuver you’re pulling
in. At a given load factor, n, (where n =
lift/weight) a level turn actually requires a higher
pitch rate than a wings-level pull-up.
For a level (constant altitude) turn at a given
velocity, pitch rate is a function of n - 1/n, but
for a wings-level pull-up it’s the smaller function
of n - 1. That greater pitch rate in the level turn
means more pitch damping. As a result a 2-g
turn, for example, requires more stick force than
a 2-g pull-up. Accordingly, a high-performance
turn takes more pilot muscle than a loop entry at
the same load factor. See the dotted versus the
solid lines in Figure 2.
Our trainers have reversible controls (wiggle an
elevator by hand and the stick wiggles as well).
In aircraft with reversible controls, at any given
altitude and c.g., the gradient of the stick forceper-g curve is independent of airspeed. Figure 3
shows how the gradient remains constant as
airspeed shifts from trim. The figure also shows
how the absolute stick force needed to obtain a
given g will depend on the relationship between
trim speed and actual airspeed. For example,
when the aircraft is flying slower than trim, static
stability leads to a nose-down pitching moment,
which adds to the pull force a pilot has to hold to
maintain a given g. But when flying faster than
trim, static stability leads to a nose-up pitching
moment that decreases the pull force necessary
to maintain a given g. Because of the change in
absolute stick force necessary to hold a given g
at speeds slower or faster than trim, test pilots try
to maintain trim speed when examining stickforce-per-g in “windup turns.” Otherwise the
data would plot an inaccurate stick force-per-g
gradient.
would mean lower forces, except that control
surface hinge moments—which are what the
pilot feels through the control system
gearing—also increase with airspeed. The
decrease in required deflection is canceled out by
the increase in hinge moment, and the stick force
required for a given g load is the same at all trim
velocities (at a constant altitude and c.g.). This
holds as long as compressibility effects
associated with high Mach numbers don’t
become a factor. Compressibility tends to
produce an increase in stick force-per-g.
Figure 3
Stick force- per-g
gradient is constant
at constant cg and
altitude.
Stick
Force,
Pull
Velocity = Trim
Velocity < Trim
Velocity > Trim
Stick
Force,
Push
g
1-g
The stick force needed to pull a given g remains
the same at any trim speed. Say the trim speed
rises. Because the elevator’s effectiveness
increases with airspeed, you don’t have to deflect
it as much to produce a given pitch rate and load
factor as you do at lower speeds. Less deflection
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6.3
Longitudinal Maneuvering Stability
Damping versus Altitude
Figure 4
Damping and
TAS
While static stability is not a function of altitude,
maneuvering stability is. Stick force-per-g goes
down as you go up. That’s because damping
decreases along with the decrease in air density
as you climb.
At least that’s the short explanation. Actually, in
responding to a given control input an airplane
doesn’t care about altitude, it cares about
airspeed. Compressibility effects aside, for a
given input it will generate the same pitching (or
rolling or yawing) moment at a given EAS
(equivalent airspeed, meaning calibrated airspeed
corrected for compressibility) regardless of
whether it’s flying down low or up high. But the
damping this moment has to overcome is a
function of altitude, because damping is a
function of TAS (true airspeed, or equivalent
airspeed corrected for density altitude), as Figure
4 explains. TAS goes up as altitude increases.
Tail arm, lT
c.g.
Pitch rate, q
ΔαΤ, change in tail angle of
attack
qlT
True Airspeed, TAS
ΔαΤ, Change in tail angle of
attack is less than above.
qlT
TAS goes up as altitude increases.
The figure shows that for a given pitch rate, q,
the velocity component generated by the
movement of the tail, qlT, is the same regardless
of altitude. But since true airspeed is higher at
altitude, the vectors add up to less change in tail
angle of attack, and so less damping.
Figure 5
Tail Volume
Coefficient
This is why an airplane will feel more responsive
and less stable at altitude, or perhaps even lower
down on a hot, high-density-altitude day. The
reduction in damping also applies to an aircraft’s
directional and lateral stability. Stability
augmentation systems, like yaw dampers, earn
their keep up high.
lT
Tail Volume
Stability depends on the restoring moment
supplied by the horizontal tail being greater than
the destabilizing moments caused by the other
parts of the aircraft. One factor is the tail-volume
coefficient, V . This is the product of the
distance between the aircraft c.g. and the tail’s
aerodynamic center, lT, times the tail area, ST.
The result is then divided by the mean
aerodynamic chord of the wing, c , times the
wing area, S.
l S
V= T T
cS
In other words, the tail volume coefficient relates
the area of the tail and its distance from the c.g.
6.4
Same tail volume
coefficient as above, but
shorter lT.
Less pitch damping makes
the aircraft more
maneuverable.
to the chord and area of the wing. It suggests
how effective the tail is going to be at producing
pitching moments. You can achieve a given tail
volume for a wing of a given size either by
having a small tail on a long fuselage, or a large
tail on a short fuselage (Figure 5).
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Longitudinal Maneuvering Stability
Since pitch damping is a function of the square
of the tail’s lever arm, lT2, the farther back your
tail is the greater the opposing aerodynamic
damping generated when you start pitching it
around to maneuver. The design criterion for
rapid maneuvering is a big tail on a short
fuselage—a hallmark of modern fighter design.
Transports have proportionately smaller tails on
longer fuselages.
Neutral Points Again
Figure 6 adds the stick-fixed maneuver neutral
point and the stick-free maneuver neutral point to
the stick-fixed and stick-free static neutral points
discussed in the ground school briefing
“Longitudinal Static Stability.” The aft shift of
the corresponding maneuver points reflects the
stabilizing effect of pitch damping. Because
damping goes down with altitude, the maneuver
points actually sneak forward as you climb.
maneuver point is the c.g. position at which stick
movement-per-g becomes zero.
If we had a weight on rails and could move the
c.g. rearward during flight, the first thing we’d
notice is a reduction in control force necessary to
change α and thus airspeed from trim (static
stability), accompanied by a reduction in stick
force needed to pull g (maneuvering stability).
Short of shifting the c.g., a knowledgeable
instructor can simulate this for a student by
manipulating the trim.
As tail volume increases, the neutral points move
aft. This in turn increases the aft c.g. loading
range.
The stick-free maneuver point is the c.g. position
at which the gradient of stick force-per-g
becomes zero. The more rearward stick-fixed
Figure 6
Stick-fixed and free
Neutral Points
←Permissible
c.g. Loading
→
Range
→
Stick-free
Maneuvering
Point: c.g. location
for zero stickforce-per-g
gradient
Stick-fixed Maneuvering
Point: c.g. location for
zero stick movement to
hold g
Aircraft c.g.
Stick-free Static
Neutral Point: c.g.
location for zero stick
force gradient to hold
airspeed from trim
Stick-fixed Static Neutral
Point: c.g. location for zero
stick movement required to
hold airspeed from trim
Typically for inherent stability and good handling qualities for an aircraft with
reversible controls, maximum permissible aft c.g. must be ahead of all static and
maneuvering neutral points, and forward of the point for minimum allowable
stick-force-per-g. Maximum forward c.g. is determined by control authority need
to raise the nose to CLmax, or by the maximum allowable stick-force-per-g.
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6.5
Longitudinal Maneuvering Stability
6.6
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